Let ∆ be the maximum degree of $G(n, p)$. Find the limit distribution of ∆, where $p = n^{−1−1/m}, m ∈ N$.
Something that I have studied regarding maximum degree for a random graph is that given $p$ to be some constant, $$P\bigl(∆> np+\sqrt{2np(1-p)\ln n} (1-x)\bigr) \rightarrow e^{-e^{-y}}$$ where $x=x(n)\rightarrow 0, y \in \mathbb{R}$.
Do we have to take the limit of this equation and find how it gives the mentioned result i.e. $e^{-e^{-y}}$ or do we need to do use this result to solve our question? I am pretty lost by the question. I understand from the limit distribution that we have to take the limit of a given equation and see if it converges to some specific constant. But I am not quite sure. Any help regarding solving the same is much appreciated.